ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 28 Feb 2016 04:45:10 +0100An issue with Root systems in sagehttps://ask.sagemath.org/question/32683/an-issue-with-root-systems-in-sage/Here is a problem that I cannot seem to understand.
<pre><code>
R=RootSystem(['A', 2])
F=R.ambient_space().fundamental_weights()
D=R.ambient_space().simple_roots()
</pre></code>
The output we get is as follows:
<pre><code>
F=Finite family {1: (1, 0, 0), 2: (1, 1, 0)}
D=Finite family {1: (1, -1, 0), 2: (0, 1, -1)}
</pre> </code>
The issue is the root lattice is contained in the weight lattice but clearly D[2] is not contained in the lattice generated by F. Perhaps I am missing something simple.
I need to write a program I would need the weyl group action on some weights and roots at the same time. How can this be done?
Thank you for your time in advance.
EDIT: Consider now the Weyl group action on weight lattice. We will continue using the notations above.
<pre><code>
W= R.ambient_space(). weyl_group()
S= W.gens()
s1, s2= S
s1.action(F[1])
</pre></code>
Gives an output
<pre> <code>
ValueError Traceback (most recent call last)
<ipython-input-12-4b0b8fd74264> in <module>()
---- 1 s1.action(F[Integer(1)])
/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/combinat/root_system/weyl_group.pyc in action(self, v)
843 """
844 if v not in self.domain():
--> 845 raise ValueError("{} is not in the domain".format(v))
846 return self.domain().from_vector(self.__matrix*v.to_vector())
847
ValueError: Lambda[1] is not in the domain
</pre> </code>
The only way I have been able to make it work is that
<pre><code>
L = R. weight_lattice()
F=L.fundamental_weights()
f= L.to_ambient_space-morphism()
s1.action(f(F[1]))
(1,0,0)
</pre></code>
This is what I can get but the problem is
<code> s2* f(F[2]) = (0,0,1). </code>
So we are in a situation of the original problem again.
I need to calculate the Weyl group action on the fundamental weights. If there is a better way to do it I would like to know. DBSSun, 28 Feb 2016 04:45:10 +0100https://ask.sagemath.org/question/32683/Error plotting root systems in Sagehttps://ask.sagemath.org/question/25415/error-plotting-root-systems-in-sage/ I am trying to use the Sage package for root systems, but it keeps returning an error. Here is what I am trying to do in the Notebook of Sage 5.7: I enter exactly the following lines from the tutorial for visualizing root systems
sage: L = RootSystem(["A",2]).ambient_space()
sage: L.plot()
The output I get is a traceback ending in 'unsupported operand parent(s) for '*': Full matrix Space of 2 by 2 dense matrices over real field with 53 bits of precision and vector space of dimension 3 over rational field'
I do not understand what I am doing wrong and what this message means. I am a newcomer to sage and all I want is to print a root system with some weights added for illustration purposes. I would appreciate any pointers.guestTue, 06 Jan 2015 22:45:41 +0100https://ask.sagemath.org/question/25415/Create an alternative: W(E8) act on positive roots to get some rootshttps://ask.sagemath.org/question/23811/create-an-alternative-we8-act-on-positive-roots-to-get-some-roots/ W=WeylGroup(['E',8])
R = RootSystem(['E',8]).root_lattice()
alpha = R.simple_roots();alpha
[w for w in W if w.action(alpha[1])==(alpha[1]+alpha[2])]
Output shows : gap: cannot extend the workspace any more!
**Is their any alternative calculation to simplify the above **BiswajitSun, 17 Aug 2014 19:13:57 +0200https://ask.sagemath.org/question/23811/epsilon basis for roots (was graph edge labels)https://ask.sagemath.org/question/9969/epsilon-basis-for-roots-was-graph-edge-labels/I need an illustration of some (sub)poset of positive roots. I have a working code that produces correct labels, but they are written as sums of simple roots. I.e. the resulting graph (when exported to LaTeX) has labels such as $\alpha_1 + \alpha_2$. I would like to have these labels in $\epsilon$-notation. I.e. the previous example would read $\epsilon_1 - \epsilon_3$.
RootSystem has an ambient_space method that provides an access epsilon basis, but
1. I am not clear on converting between these two bases
and
2. output to LaTeX should really use $\epsilon_1 - \epsilon_3$ rather than $(1,0,-1)$.vit.tucekMon, 01 Apr 2013 11:43:19 +0200https://ask.sagemath.org/question/9969/Producing subgroups of Weyl groupshttps://ask.sagemath.org/question/8606/producing-subgroups-of-weyl-groups/Let W be a Weyl group, e.g.
W = RootSystem('[A, 4]').weight_lattice().weyl_group
Given some elements $S \subset W$, I would like to produce the subgroup generated by $S$. It seems like there are methods in SAGE to do this when W is an abstract group, but I can't see how to do it when $W$ is a Weyl group. Any suggestions?markblunkWed, 04 Jan 2012 15:42:04 +0100https://ask.sagemath.org/question/8606/dual of weyl grouphttps://ask.sagemath.org/question/8217/dual-of-weyl-group/I was wondering if it is possible to, given an element in a Weyl group, produce the corresponding element in the dual Weyl group. As an example, if
`w` in `W = RootSystem(['A', 3]).weight_lattice().weyl_group()`
Then I would like a function f such that
`f(w)` in `RootSystem(['A', 3]).coroot_lattice().weyl_group()`,
with the obvious duality
`<w*x,y> = <x,f(w)*(y)>`,
where x in the weight lattice and y is in the coroot lattice.
thanksmarkblunkSat, 09 Jul 2011 18:29:38 +0200https://ask.sagemath.org/question/8217/Dictating the roots to RootSystemhttps://ask.sagemath.org/question/7928/dictating-the-roots-to-rootsystem/Is there a way to dictate the choice of roots/simple roots to the RootSystem command?hoylandSun, 13 Feb 2011 18:32:19 +0100https://ask.sagemath.org/question/7928/